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Linear Quadratic Optimal Control for Systems Governed by First-Order Hyperbolic Partial Differential Equations

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Journal of Systems Science & Complexity 2024年第1期37卷 230-252页

作者： XUE Xiaomin XU Juanjuan zhang huanshui School of Control Science and Engineering Shandong UniversityJinan 250061China College of Electrical Engineering and Automation Shandong University of Science and TechnologyQingdao 266000China

This paper focuses on linear-quadratic(LQ)optimal control for a class of systems governed by first-order hyperbolic partial differential equations(PDEs).Different from most of the previous works,an approach of discretization-then-continuousization is proposed in this paper to cope with the infinite-dimensional nature of PDE systems.The contributions of this paper consist of the following aspects:(1)The differential Riccati equations and the solvability condition of the LQ optimal control problems are obtained via the discretization-then-continuousization method.(2)A numerical calculation way of the differential Riccati equations and a practical design way of the optimal controller are proposed.Meanwhile,the relationship between the optimal costate and the optimal state is established by solving a set of forward and backward partial difference equations(FBPDEs).(3)The correctness of the method used in this paper is verified by a complementary continuous method and the comparative analysis with the existing operator results is presented.It is shown that the proposed results not only contain the classic results of the standard LQ control problem of systems governed by ordinary differential equations as a special case,but also support the existing operator results and give a more convenient form of computation.

关键词： Discretization-then-continuousization method first-order hyperbolic partial differential equations forward and backward partial difference equations linear quadratic optimal control.

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Explicit solution to forward and backward stochastic differential equations with state delay and its application to optimal control

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Control Theory and Technology 2022年第3期20卷 303-315页

作者： Tianfu Ma Juanjuan Xu huanshui zhang Control Science and Engineering Shandong UniversityJinan250061ShandongChina College of Electrical Engineering and Automation Shandong University of Science and TechnologyQingdao266590ShandongChina

This paper is concerned with linear forward–backward stochastic differential equations(FBSDEs)with state delay,the solvability which is much more complex than the case of no delay or input delay caused by the prediction of the backward processes of the future time.To overcome this difficulty,we innovatively establish the non-homogeneous relationship between the backward and forward processes with the help of the corresponding discrete-time system.The main contribution is to give the explicit solution to the FBSDEs with state delay in terms of partial Riccati equations for the first time.The presented results form the basis to solve the challenging problem of linear quadratic optimal control for multiplicative-noise stochastic systems with state delay.

关键词： FBSDEs State delay Analytical solution Discrete-time system

Optimal Kalman-like filter for a class of nonlinear stochastic systems

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Journal of Ocean Engineering and Science 2023年第5期8卷 500-507页

作者： Shulan Kong Yawen Sun huanshui zhang School of Mathematics Science Qufu Normal UniversityQufuChina College of Electrical Engineering and Automation Shandong University of Science and TechnologyQingdaoChina

This paper deals with an optimal Kalman-like filter for nonlinear discrete-time systems aided with auto and cross-correlated noises and stochastic parameter matrices involved in state and measurement equa-tions,and random nonlinearity.The random variables are proposed by their statistical characteristics while the inquiry is focused on stochastic multivariate analysis and calculation.For the nonlinear sys-tem with the auto and cross-correlated noises and stochastic parameter matrices,an equivalent system is first reconstructed by decomposing stochastic parameter matrices and introducing uncorrelated pseudo-noises.Then a recursive filter that ensures unbiasedness and minimizes the error variance is designed for the newly transformed equivalent system.Finally,the filter is verified by applying it to some numerical simulations.

关键词： Kalman-like filter Multivariate analysis Auto-correlated and cross-correlated stochastic parameters Uncorrelated pseudo-noises

Time-inconsistent stochastic linear quadratic control for discrete-time systems

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Science China(Information Sciences) 2017年第12期60卷 44-56页

作者： Qingyuan QI huanshui zhang School of Control Science and Engineering Shandong University

This paper is mainly concerned with the time-inconsistent stochastic linear quadratic(LQ) control problem in a more general formulation for discrete-time systems. The time-inconsistency arises from three aspects: the coefficient matrices depending on the initial pair, the terminal of the cost function involving the initial pair together with the nonlinear terms of the conditional expectation. The main contributions are: firstly,the maximum principle is derived by using variational methods, which forms a flow of forward and backward stochastic difference equations(FBSDE); secondly, in the case of the system state being one-dimensional, the equilibrium control is obtained by solving the FBSDE with feedback gain based on several nonsymmetric Riccati equations; finally, the necessary and sufficient solvability condition for the time-inconsistent LQ control problem is presented explicitly. The key techniques adopted are the maximum principle and the solution to the FBSDE developed in this paper.

关键词： time-inconsistency open-loop equilibrium control maximum principle FBSDE nonsymmetric Riccati equations

Optimal control with irregular performance

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Science China(Information Sciences) 2019年第9期62卷 116-129页

作者： huanshui zhang Juanjuan XU School of Control Science and Engineering Shandong University

In this paper, we solve the long-standing fundamental problem of irregular linear-quadratic（LQ）optimal control, which has received significant attention since the 1960s. We derive the optimal controllers via the key technique of finding the analytical solutions to two different forward and backward differential equations（FBDEs）. We give a complete solution to the finite-horizon irregular LQ control problem using a new ‘two-layer optimization’ approach. We also obtain the necessary and sufficient condition for the existence of optimal and stabilizing solutions in the infinite-horizon case in terms of solutions to two Riccati equations and the stabilization of one specific system. For the first time, we explore the essential differences between irregular and standard LQ control, making a fundamental contribution to classical LQ control theory.We show that irregular LQ control is totally different from regular control as the irregular controller must guarantee the terminal state constraint of P;（T）x（T） = 0.

关键词： irregular LQ control Riccati equation stabilization

Achievable delay margin using LTI control for plants with unstable complex poles

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Science China(Information Sciences) 2018年第9期61卷 182-189页

作者： Peijun JU huanshui zhang School of Control Science and Engineering Shandong University School of Mathematics and Statistics Taishan University

We consider the achievable delay margin of a real rational and strictly proper plant, with unstable complex poles, by a linear time-invariant(LTI) controller. The delay margin is defined as the largest time delay such that, for any delay less than this value, the closed-loop stability is maintained. Drawing upon a frequency domain method, particularly a bilinear transform technique, we provide an upper bound of the delay margin, which requires computing the maximum of a one-variable function. Finally, the effectiveness of the theoretical results is demonstrated through a numerical example.

关键词： delay margin systems with time-delay time-invariant systems unstable complex poles frequency domain method

Stabilization analysis for Markov jump systems with multiplicative noise and indefinite weight costs

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Science China(Information Sciences) 2021年第5期64卷 111-120页

作者： Hongdan LI Chunyan HAN huanshui zhang School of Control Science and Engineering Shandong University School of Electrical Engineering University of Jinan

This paper mainly discusses the stabilization problem for discrete-time Markov jump linear systems(MJLSs) involving multiplicative noise with an infinite horizon. The cost weighting matrices are generalized to be indefinite. To the best of our knowledge, this paper is novel and unlike most previous studies, it provides the necessary and sufficient conditions that stabilize the MJLSs in the mean square sense with indefinite weighting matrices.

关键词： stabilization indefinite multiplicative noise Markov jump linear system

A leader-follower stochastic linear quadratic differential game with time delay

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Science China(Information Sciences) 2018年第11期61卷 86-98页

作者： Juanjuan XU Jingtao SHI huanshui zhang School of Control Science and Engineering Shandong University School of Mathematics Shandong University

In this paper, we are concerned with the leader-follower stochastic differential game of It?o type with time delay appearing in the leader’s control. The open-loop solution is explicitly given in the form of the conditional expectation with respect to several symmetric Riccati equations. The key technique is to establish the nonhomogeneous relationship between the forward variables and the backward ones obtained in the optimization problems of both the follower and the leader.

关键词： time delay leader-follower differential game open-loop strategy Riccati equation

Consensus for high-order multi-agent systems with communication delay

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Science China(Information Sciences) 2017年第9期60卷 241-252页

作者： Zhenhua WANG huanshui zhang Minyue FU Huaxiang zhang School of Information Science and Engineering Shandong Normal University Institute of Data Science and Technology Shandong Normal University School of Control Science and Engineering Shandong University School of Electrical Engineering and Computer Science University of Newcastle School of Automation Guangdong Key Laboratory of Io T Information TechnologyGuangdong University of Technology

In this study, consensus problem for general high-order multi-agent systems with communication delay is investigated. Given the unstable agent dynamics and a known communication delay, two consensus protocols are designed to guarantee consensus over undirected network. By jointly researching the effects of agent dynamics and network topology, allowable delay bounds depending on the maxima of concave functions are easy to calculate. Especially, the maximum delay bound is derived when the network topology is completely connected. The main approach for the same involves designing the control gains on the basis of the solution of a parametric algebraic Riccati equation. Finally, the theoretical results are demonstrated via numerical simulations.

关键词： consensus communication delay historical input information parametric algebraic Riccati equa tion eigenratio